The "floor" is due to solar neutrinos - the sun is a continuous source of neutrinos and is located much closer to us than virtually every past supernova was. It is also a soft floor, as there are analysis tricks that can be played to distinguish between interactions from neutrinos and WIMPs at the statistical level. For this reason, the current fashion is to refer to a "neutrino fog," which can be entered into to some depth, as opposed to a "neutrino floor."
This is absolutely correct, as there are no solar neutrinos with energy above ~20 MeV (but below that, the solars dominate). Thanks for the clarification.
The interesting thing is the physical process creating the floor is a fairly novel one, coherent scattering of neutrinos off nuclei. This is where the neutrino scatters off all the nucleons together, with the contributions adding coherently, rather than incoherent scattering off individual nucleons.
This process was first observed only fairly recently, at the Spallation Neutron Source using neutrinos from stopped pions. IIRC, they realized a service corridor under the target room would be unintentionally suitable for the experiment.
